Evaluate the following surface integral $I$ $$ \begin{aligned} &I=\iint\limits_S f(x;y;z)\ dS\ \ \text{where}\\ &\text{$S$ is an ellipsoid}\ \ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\ \ \text{and}\ \ f=\sqrt{\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}} \end{aligned} $$
I used a substitution for $x, y$ and $z$.
$$
\begin{aligned}
\begin{cases}
x=a\sin\theta\cos\phi\\
y=b\sin\theta\sin\phi\\
z=c\cos\theta
\end{cases},\ \ \ 0\leqslant\theta\leqslant\pi,\ \ 0\leqslant\phi<2\pi
\end{aligned}
$$
Then, I tried to calculate $dS$.
$$
\begin{aligned}
&r=r(x,y,z),\ \ dS=\sqrt{|G|}\ d\phi d\theta,\ \ \text{where $G$ is a Gramian matrix}\\
&G=\begin{pmatrix}
(r'_\phi,r'_\phi) & (r'_\phi,r'_\theta)\\
(r'_\theta,r'_\phi) & (r'_\theta,r'_\theta)
\end{pmatrix}
\end{aligned}
$$
However, the calculations of $G$ became very tough, and now I doubt that it was the right way to solve this problem.
So, the question is how should I approach this problem?
